Elliptic operator

Elliptic partial differential equations are a special class of partial differential equations ( PDE ). They are formulated with the help of elliptic differential operators. The solutions of an elliptic partial differential equation have certain properties, which are explained in more detail here. The Laplacian is the most well-known elliptic differential operator, and the Poisson equation is the corresponding partial differential equation.

• 5.1 existential statement
• 5.2 regularity
• 5.3 Maximum Principle
• 5.4 Eigenvalue Problems

• 6.1 Definition
• 6.2 invertibility
• 6.3 Singular carrier

Physical interpretation

The elliptic equation is a generalization of the Laplace equation, and the Poisson equation. An elliptical second order differential equation in the form of

Which must satisfy the coefficient functions, and suitable conditions.

Such differential equations typically occur in the context of stationary ( time-independent ) problems. They often describe a state of minimum energy. The said Laplace and Poisson equations describe about the temperature distribution in a body or the electrostatic charge distribution in a body. Other elliptic equations are used for example for studying the concentration of certain chemical substances. The terms of order two here describe the diffusion. The first-order terms describe the transport, and the term of order zero describes the local increase and decrease.

Non-linear elliptic differential equations occur also on the calculus of variations and differential geometry.

Definition

Elliptic differential operator

A differential operator, recorded in multi- index notation, the order in an area called the elliptic point if

Is satisfied for all. It is called the principal symbol of. A differential operator is called elliptic if it is elliptic for all.

Elliptic differential equation

Be an elliptic differential operator and a function, then that means the equation

Elliptic differential equation and is the unknown function in this equation.

Uniformly elliptic differential operator

A differential operator is called uniformly elliptic in if there is one, so that

Applies to all.

Hypo- elliptic differential operator

An operator with constant coefficients is called hypo -elliptic if there is a such that for all and with all the following applies:

• And
• .

General ie, a differential operator on an open set with not necessarily constant coefficients hypo- elliptic, if it is open for any amount, limited and each distribution, the implication

Applies. In words: If the image is in the distribution sense of the differential operator infinitely differentiable, as is already the case for the pre-images.

In contrast to the uniformly elliptic differential operator of the hypo- elliptic differential operator is a generalization of the elliptic differential operator. This exposure to the differential operator is thus weaker. See the regularity theory of elliptic operators below.

Origin of the name

The adjective elliptical in term elliptic partial differential equation derived from the theory of conic sections. In this theory, in the case the amount of solution of the equation

Called ellipse. Referring now to the homogeneous differential equation

Second order in two dimensions with constant coefficients, so this is exactly then uniformly elliptic if the following applies.

Examples

• The most important example of a uniformly elliptic differential operator is the Laplacian

• The Cauchy -Riemann operator

• The parabolic partial differential operator is hypo -elliptic, but not uniformly elliptic. The parabolic differential equation is called the heat equation.

Theory of elliptic differential equations of second order

The following are the key messages for elliptic differential operators of order two are shown in n dimensions. Be therefore

An elliptic differential operator of order two. Furthermore, it is an open, connected, bounded subset with Lipschitz regular boundary.

Existence theorem

Let the coefficient functions are all measurable and bounded functions. Then for each a unique weak solution of the Dirichlet boundary value problem

If the differential operator associated to the bilinear form is coercive. This is defined by virtue of

By Lemma of Lax- Milgram we deduce the existence and uniqueness of the solution of the bilinear form. Is uniformly elliptic, then the associated bilinear form is always coercive. One instead of a Dirichlet boundary condition used a Neumann boundary condition, then there exists if the associated bilinear form is coercive again, exactly one solution of the partial differential equation, which can be proved almost the same.

Regularity

Be prepared for all, and was well, and a weak solution of the elliptic differential equation

Then we have.

Maximum principle

For elliptic differential operators of second order maximum principle applies. Be in and be.

1 If

Applies and a non- negative maximum at an interior point of accepting, then is constant.

2 If

And applies a non- positive minimum at an interior point of assuming, then constant.

Eigenvalue problems

Consider the boundary value problem

With an eigenvalue of the differential operator is. It should also be symmetric differential operator.

1 Then all the eigenvalues ​​are real.

2 In addition, all eigenvalues ​​have the same sign and have finite multiplicity.

3 Finally, there exists an orthonormal basis of with the eigenfunction corresponding to the eigenvalue.

Theory of elliptic pseudo-differential operators

Definition

A pseudo- differential operator is called elliptic if its symbol is actually carried and the homogeneous principal symbol is uniformly elliptic - or equivalently, if in a conical neighborhood of the real symbol for a constant for the inequality and apply.

Invertibility

Be an elliptic pseudo- differential operator and, then there exists a pseudo- differential operator actually worn so that

Applies. It is the identity operator, and is an operator which maps each distribution on a smooth function. This operator is called parametrix. The operator can thus be inverted modulo. This property makes the elliptic pseudo- differential operator and thus. As a special case of the elliptic differential operator to a Fredholm operator

Singular carrier

Be again an elliptic pseudo- differential operator and. Then, for any distribution

The singular support of a distribution that is not changing.